Computing-table.



J. KN AUSB.

GOMPGTING TABLE.

APPLICATION 111.211 1120.1, 1911.

Patented Sept; 3, 1,912.

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COMPUTING TABLE.

APPLIOATION FILED DB0. 1, 1911 Patented Sept. 3, 1912.

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COMPUTING TABLE.

APPLICATION FILED 12110.), i911. t u

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J. KNAUSE.

COMPUTING TABLE.

APPLIGATION ULEB mm1, 1011 Joh/n/ Knaus@ .T OHN RNaUsE, oF LocUsT CORNER, oHIo, AssIGNoR or ONE-HALF :ro HAYs, or Locus'r CORNER, oHIo.

UNITED i STATES PATENT OFFICE.

oHAR1 .Rsl o.

CoMPU'rING-TARLE.

- Specification of ietters Patent.

Patented 'sepa-s, 1912,.

Appi'ication fried December i, 191.1.. seriai No. 663,407.

ein it tlieUnited States, residing at Loexi'ifs't rrier, in' tliecounty of Clermont and State ,of`l Ohio,l have invented certain ne1 and useful Improvements in Computing- Tables, of which the following is a speci- Y Iieation.

Ihe invention -relates toa tabular calcuf ceeding tivo figures, for which the particu- Vlar table is' constructed;4 and the object of the inventionvisl to provide a computing t tableby means .of which vsuch lmultiplicatiorrcr division can generally beaccom Aplished by a mere inspectiono'f the table.' y L ;In ,tliefaccompanying drawings, f orming part of" thisv specification, Figure 1- shows the beginni'iig'of la'talole constructed .in ac-l cordance witlithe present invention'ior multiplying :by a. mere inspection any.l number:- not exceeding'four figures bythe nuinf bei-1,. 2341s a multiplier, or for dividing any number not greater than 229,977, by the number` 23 as a divisor; Fig. 2 shows the firstpart or4 half of a computing table constructed according to the present invention for multiplying by inspection any number not exceeding tour igures by thev number 89 as a iuciliplieruor for dividing any number not'gi'cater than 889,911 by the number 89 a. divisor, Fig. 3. a continuation showingtl-,c second pai-t -or half of the \able;.and Fig. si shows the Iirst part of a modified or extended forni of a computing table con striited according to the present invention foi'"miiltiplying by inspection any number notexceeding six figures by the number 99 as a multiplier, or for dividing any number not eirceedilng 98,999,901 by the number 99'asa divisor.

lnfthepreferred -or simplen form of -tlie middle column of which is designated for distinction by the letter X placed atthe head of the column, and contains in series down the column numbers corresponding" to the two intermediate ligui'es at ziiiy'pronamely, O, l, 2, 3, 4:. *senting respectively the riglithand oriinit U' posed multiplicand, that is to say, the 'numf I Be it known that I, JOHN KNAUSE, a citierals indicating the ten and hundred -figures thereof. In'this column the saine. number 1s occasionally duplicated, asshown, for the particular purposes of the table;v and cipliei's are prefixed when necessary to make Y a two-figure number. .f

To the left of the middle or X column are provided ten several columns forming what is designated as Division A at the head of these columns, and the head of each column is further designated by .the teniarabic digits or numerals, namely,0, l, 2, 3, 4,5, 6, 7 8 and 9, representing respectively the lefthand or thousand ligure of any proposed multiplicand, so .that this division might well be called the thousand division of the table And to the right of the middle or X column are also provided ten several `columns forming what is designated as Division, atthe bead of these columns, and the 4head of each column is fui-ther desig- -nated bythe ten' arabic digits or numerals,

i. 6,' 7, 8 and 9, reprefigure of any proposed multiplicand, so that this division might nell be called the unitA division of the table.

The several vertical columns are preferably sepa'rated for convenience of iiispec-' .tion by the vertical lines aand b', the vertiv ca l lines a. dividing the several columns in "Divi, ion A and Division B being preferably distinguished frein-the vertical lines() at cach side of the middle or X column and at each side of the table, which distinction -may bef made by using a single line foi-the former and a double line .for the latter, as shown, or' by printing the respective lines in different widths or colors.

In the construction of the table, consecu- `tive multiples of the multiplier to be used in' a given table are entered in series from left to 'right in the several rows of Division B.

These `'multiples begin `vithOOO in'rthe first row of the O column and thence inseries to 'the right are'entered the successive -multiples as 023, O46, 069,-etc., as shown inFig l for the multiplier 23; and so on -i'n the severaLsuccessive 'rows' up. to t-lielaigest multiple .which 'can be expressed' in' three` figures 989 located in row' in theicolumn, and column 3 in Division B. In this printing the respect-ive lines' in diierent division, ciphers are'prefix'ed to fthe smallerA multiples when necessary to make a-'threefigure-numbers. This multiple'and row completes what ma be termed and is designated Section 1 at eit er side of the table shown in part in Fig. 1, and the next succeeding multiple 1012 of the multiplier' instead of being entered in the table iii the same transverse r'ow with the 989.miiltiple,- is entered in the next row below; andthe thousand figure 1 is omitted, so that this multiple appears in column 4 of Division B as 012, being the first multiple in the first row of Section 2 of the table, in which row the'nuinber 04 of the X column is repeated as the firstl numbei' in Section 2.' Beginning With the multiple (1)012 in Section 2'of Division B, consecutive multiples of the multiplier are entered in series from leftto right and in successive rows as before, until the next thousand is reached, 'atwhich We have the multiple (1)978 entered in ro'w 08 in the middle or X column and in-coluiiin 6 of Division B, tliis'number being the last number in Section 2; and the last three figures of the next succeeding multiple (2)001 are entered in column 7 of Division B in the next row below and become the firstmultiple in Section 3 of the table, for which row the number 08 is duplicated or repeated in the middle or X column of the table. l

The several transverse rows are preferably separated for convenience of inspection by the transverse lines c and d, the transverse lines c dividing the several rows in each section being preferably distinguished from the transverse lines d between the several sections,m'hicl,i distinction may by using a lighter line for the formerand a heavier line for the latter, as shown, or by colors. Y A

In -the thousand vDivision A of the simpler form of table, only one row of multiples of the multiplier is entered and used in eiicli section of the table.v In Section 1, the niultiple begins with 0 in the cipher column, and thence in series to the right are entered the successive multiples A23, 46, 69, and so on to 207 lin column 9 of this division; and, for

' vthe purposes of the table, it is not necessary to prefix cipliers to-the respective multiples to make a three-figure number as is done in Division B of .the table. In the next section in Division A of the table, tlie row of numbers' i's each increased by one so that we have the series 1, 24, 47, and 'so onto 208 in Sec- Y tion 2; and the numbers 2, 25, 48, and so on to 209 in Section 3; and likewise .for each. succeeding section of-the table..

in the use 'of the able, ums, described, whichis constructed for the multiplier 23. fnr'multiplyiiig any number not-exceeding be made four figures, as, `for instance, 3147, the multiplicaiid is pointed off or separated mentally or otherwise into the thousand figure 3, the tivo intermediate figures 14 and the unit figure 7, as follows-.43 14 7; and the thr'e'e righthand figures of the product arereadiy vfound by mere inspection in column 7 ci' Division B and in the row designated Min the X column-the figures thus found table being 381; and the,lefthand"figures the product are readily found by mere iiispection in column 3 of Division A inthe sanie Sect-ion 4 wherein the right-hand figures were ,found-which left-'hand figures as shown by the table are 72: and by merely prefixing these left-hand figures to the iight-liand figures, we have tlie proper product 72,381, from multiplying 3147 by 23.

From the foregoing description of the construction anjd use of the table, it appears l that the tivo-figure numbers in the middle or X4 column are continued consecutively exj cept when a sect-ion ends before the last row of multiples in Division B is completed, that is, before all the columns are filled; in which event the. number in the X column must 'be continuedgcir repeated until such line is completed, one row below in the first row of the next section, since 'the' several multiples in the broken-line in Division B require the same number-in the X.column, butncorrespond to the numbers in Division Anfwhich are entered in the respective sections in Division B.

A complete table for multiplying by a mere inspection any number containing not m'ore than four figures by the number 89 as a multi lier is shown in two parts in Figs. 2

and 3,. ig. 2 being part 1 and Fig. 3 being.

part 2 of the table, and part- 2 being merely a eontinuationof part 1. In this table it will be seen that the several sections, (which are not designated' excepting as they are lseparated bylieavier transverse lines), are

limited to tivo parallel transverse rows -in Division B. of the table; and it is evident enerally `versely with the value of the multiplier, and that the number of-sections in a'given table is equal to the number of units iii-the multilier. i D p In the useof the table illustrated in 2 and 3 for 'multiplying'by mere inspection any number containing not more than four figures, as, for instance, 7859, the product 'cangfbe readily obtained by merely separatthat the size of sections varies in- A A i th'emultiplicand as before described, .as

fo ows( 85 9; after which the three right figures ofthe productf are readily found in column 9 of Division B and row 85- as designated in tlie X column,v the three figures thus foundbeing 451; to which are prefixed the or number found in column 'I of Divisxfn. A'in the sam'eection from which the right-hand figures are taken, the prefix figures thus found being 699; thus making the complete product (399,451. In the use of the same table, if it be dcsired''to multiply any number consisting of more 'than four figures by the number 89, the same can be done by beginning at the right and dividing the multiplicand into periods of four figures each, as, for instance,'if the multiplicand is 2,785,349, it is first divided as follows: 278 5349, whereupon the multiplication of the first four figures, 53h19, by S9 is made as before described by van inspection of the table and a lpartial productI of 476,001 is found; after which the multiplication of the next three figures, 278, (with a cipher prefixed to make four figures 0278), is likewise made by a mere inspection of the table and a second partial product is found to be 24,742, which, being located four places to the left and added to tbe first partial product, gives the .complete product of 247,896,061 for the multiplication.

These examples illustrate the general use of computing tables constructed, as described above, according to the present invention; and general rules may be stated as follows lfor making any multiplication.

Multiplication.

ber thus found is prefixed to the right-hand figures already I found and the product is thus completed.

Oase IL To multiply any number consisting of five or more figures by any other number not exceeding two figures.

Rule- Beginning at the right, divide the multiplicand into periods of four places each. Proceed with the four figures in the first period as directed in Case I for the first partial product. Then proceed with the figures in the next eriod in the same manner for the second pa'rtial product and write the same four places to the leftunder the first partial product. Proceed with the figures in each additional period in the same manner, always writing each successive partial product four places to the left of its predecessor; and the sum of the several partial products thus Written is the complete product.

Oase I I I .-lVhen multiplier consists of three or more figures.

Hula-Beginning at the right, divide the evident that, by reversing the processes,

the tables may he used for division as well as multiplication. the table constructed for a particular multiplier being employed for the same number as a divisor; and for such use the normal capacity of the table is a dividend equal to 9,999 multiplied by the multiplier for which the particular table is constructed, as, for instance, 889,911 in the table constructed for 89 as the multiplier; General rules may be stated as follows for such Divison.

Oase [.Wvhen the dividend. is the capacity of the table or less.

Ruler-Point off three figures from right of dividend. Find. remaining figures, or nearest J'less in first row of Division A, and proceed downward in the same column until exact figures are found. The figure heading the column so found is first figure of the quotient. in Division B in the .same section find the three figures pointed off, or the nearest number less. In the same row of column X find the next two figures of the quotient, and, at the head of the same vertical column, nd the fourth figure of the quotient. If the number referred to is less than the three figures of the dividend pointed off', subtract the 011e from the other and the difference will be the remainder.v

("ase II.- Vhen dividend is greater than the capacity of the table. Begin at right of dividend and point off' three figures. or as man v times three figures as may be necessary to reduce dividend to capacity of table. Then proceed with the left-hand period as in Case I. In the event a remainder is left. annex the next period to the remainder and proceed as before for each successive period until operation iscomplete.

las@ IIL- To reduce a. remainder to a decimal.

[hda-Annex one cipher to remainder. Proceed as in Case I to find a number one less than remainder with annexed cipher. The figure. at the head of the column so found is first figure of decimal. In the same transverse section in Division B, find greatest number. The figures in the same row in the X columnare the next twof decimal places, and the egure at the head of the column containing said greatest number is the fourth decimal place.

Noten-Should dividend contain fewer than four figures, prefix ciphers to make four places. Very rarely, the first period of the dividend is less than the smallestl number inv the same sect-ion; in which case, use the preceding section, and-then subtract the largest number in Division B, in the same sect-ion, from the first period with 1 prefixed.

A modified or extended form of table constructed according to the present invention is illustrated in Fig. 4. The principle of construction in this extended form is substantially the same as for the simpler form, but admits the use of a mnltiplicand of six figures and the capacit-y of a section is ten thousand instead of one thousand. The columns in Division B contain consecutive multiples of the multiplier up to the largest which can be expressed in four figures, and the numbers in Section lof the ten thousand-Division A are the same as in Section B except the prefixed ciphers are omitted in `the multiples containing less than four figures and the multiples occurring in the last row of the section are also omitted. In Section 2 and in each succeeding section of Division A, the numbers are respectively greater by a unit than the corresponding number in the preceding section; so that each section of Division A has always ,just ten rows of multiples. The intermediate column includes a series of numbers each composed of three figures from 000 up to 999 in a complete table (not shown), the number in'the X column in the row at the bottom of each section being repeated in the row at the top of the Section next below. And finally, an additional column is provided at thc left of the table in which are entered the numerals 0, 1, 2, 3, 5, (i, 7, 8 and 9, respectively. in the first ten rows .of each section. which additional column ma)v be termed the hundred thousand column as the figures in each section correspond with the first figure of any particular multiplicand having six figures. For the use of this extended table, the rule `for multiplication is practically the same as explained above for the simpler table, and may jbe expressed as follows:

?'v/e.-l `ind first or unit figure of multiplicand at the head of Division B and proceed downward in the same unit column containing this figure to the row containing I the next three figures of the multiplicand in the X column,'in which row and in the same unit colmun will be found thc first four figures of product. Find the fifth figure of the multiplicand at the head of the column in the ten thousand Division A and proceed downward to the same section from Which'row and ten thousand column will be found the remainder of product to be prefixed to the partial product already found vin the unit division.

vI claim:

l. A computing table including a middle column containing a series of numbers severally representing the intermediate figures of a multiplicand, a lateral division containing a series of columns having therein a series of multiples of a particular mult-iplier severally representing the right-hand figures of a product. of a multiplicand by the multiplier, and another lateral division containing a series of columns having therein av series of numbers severally representing the left-hand figures of said product.

2. A computing table including a middle column containing a series of numbers severally representing the intermediate figures of a multiplicand, a lateral division having ten columns headed by numerals in series severally representing the unit figure of the multiplicand, and another lateral division having ten columns headed by numerals in series severally representing a left-hand figure of the multiplicand; there being multiples of a particular multiplier entered in series and transverse rows in the columns of one division, ciphers being prefixed or the left-hand figure omitted when necessary to make a given-figure number, and the first multiple from which each several left-hand figure is omitted being entered in a row below the last preceding multiple to form,-l

separate transverse sections, the last number in the middle column in each section being repeated as the first number in the next section when a row of multiples is broken; and there being a series of numbers in the first section of the other division like the first row of multiples in the one division, and a similar series of numbers cach increased by one unit in each succeeding section of said other division.

3. A computing table composed of a middle column containing a series of numbers severally representing the two intermediate figures of a four-figure multiplicand. a lateral division containing ten columns headed by numerals in series severally representing the unit figure of the multiplicand, and another lateral division having ten columns headed by numerals in series severally representing the thousand figure of the multiplicand; 4there being a series of multiples ofa particular multiplier/entered 1n serles and transverse rows in the columns of Aone division, ciphers being prefixed or the lefthand figure omitted when necessary to make a three-figure number, and the first multiple l first section of the other division like the 10 from which each several left-hand ligure is omitted bing entered in ay row below J(he last preceding multiple to form separate transverse sections,the last number in the middle column in each section being repeated as the first number in the next section when a. row of multiples is broken; and there being a series of numbers in .the

irst row of multiples in the onel division, and a similar series of numbers each increased by one unit in each succeeding section of said other division.

JOHN KNAUSE. Witnesses:

W. A. J osEPH, T. R. BREEDING. 

